Question

to pass a test you have to perform successfully two consecutive tasks, one easy and one hard. the easy task you think you can perform with probability z.and the hard task you think you can perform with probability h, where h<z. you are allowed three attempts. either in the order(easy, hard, easy)or in the order (hard, easy, hard). whichever order, you must be successful twice in a row to pas. assuming that your attempts are independent, in what order should you choose to talk to take the tasks in order to maximize your probability of passing the test?

Answer 1

Here the major point is to note that the trials are independent and you need to be successful twice in a row to pass.

H denote hard task and E denote east task. P[H]=h, P[E]=z and also z<h mea ing easier task is have more probability to be successful in .

But the thing to note here is that in order H E H tasks we need to pass either H followed by E or E followed by H. In second case order E H E we need to pass E followed by H or H followed by E.

All these probabilities are equal and is z* h or h*z since they are independent. So does these mean the order doesnt matter?

The ordet still matters because here we considered only passing in first two tasks or second two tasks in order. We need to also consider the AND probabiliry.

For order EHE

Let A denote E followed by H

B denote H followed by E

P[ A or B] = P[A] + P[ B] - P[ A and B]

= zh+hz-zhz

Similarly if we define for order HEH

C as H followed by E and D as E followed by H

P[ C or D]= hz + zh - hzh

Since zhz > hzh since z> h

Obviosuly Probability of C or D happening is greater.

So choose order HZH for best probability of passing.